A subdifferential characterization of Motzkin decomposable functions J. E. Martı́nez-Legaz Departament d’Economia i d’Història Econòmica Universitat Autònoma de Barcelona and Barcelona Graduate School of Mathematics (BGSMath) Spain L. Thibault Institut Montpelliérain Alexander Grothendieck Université de Montpellier France May 24, 2017 Abstract The paper provides a new subdifferential characterization for Motzkin decomposable (convex) functions. This characterization leads to diverse stability properties for such a decomposability for operations like addition and composition. Keywords: Motzkin decomposition, recession cone, subdifferential, Legendre- Fenchel conjugate. 1 Introduction T. Motzkin proved in his 1936 paper [5] that any polyhedral convex set in Rn can be written as the (Minkowski) sum of a polytope and a polyhedral convex cone. Based on this result, subsets in Rn which can be decomposed as the sum of a convex compact set and a closed convex cone have been studied in great detail in [1], where the name of Motzkin decomposable have been coined for such sets. Various characterizations and properties of Motzkin decomposable sets are established in [1, 2, 3, 4]. In [1] Motzkin decomposable functions are introduced as functions whose epigraphs are Motzkin decomposable sets, and this paper puts in light many properties of such functions. The aim of the present paper is to provide a new characterization of Motzkin decomposable functions via their subdifferentials and to derive certain stability results related to some operations. In Section 2 we recall diverse basic notions 1 and features in Convex Analysis which are utilized through the paper. Section 3 is devoted to the subdifferential characterization of the Motzkin decomposability for convex functions. In the last section we prove the results on the stability of this property for functions, when operations like addition and composition are concerned. 2 Preliminaries Throughout h·, ·i will denote the usual Euclidean inner product on Rn and k · k the Euclidean associated norm, and B will stand for the closed unit ball centered at the origin. If S is a subset of Rn , its closure (resp. convex hull) will be denoted by cl S (resp. co S). Given an extended real-valued function f : Rn → R ∪ {−∞, +∞}, we recall that its (effective) domain dom f and its epigraph epi f are the sets dom f := {x ∈ Rn : f (x) < +∞} and epi f := {(x, α) ∈ Rn × R : f (x) ≤ α}. The function f is said to be proper whenever it is finite at some point and does not take the value −∞. The Legendre-Fenchel conjugate f ∗ : Rn → R ∪ {−∞, +∞} is defined by  f ∗ (x∗ ) := sup hx, x∗ i − f (x) for all x∗ ∈ Rn , x∈Rn and the supremum can be equivalently taken over x ∈ dom f . Clearly, f ∗ is a lower semicontinuous (lsc, for short) convex function. Further, when f is a proper lsc convex function, it is well-known that so is f ∗ and (f ∗ )∗ = f . If the function f is convex, its subdifferential ∂f (x) at a point x where it is finite is the set ∂f (x) := {x∗ ∈ Rn : hy − x, x∗ i ≤ f (y) − f (x), ∀y ∈ Rn }, or otherwise stated x∗ ∈ ∂f (x) ⇔ hx, x∗ i = f (x) + f ∗ (x∗ ). (1) So, when f is in addition proper and lsc, one sees that x∗ ∈ ∂f (x) ⇔ x ∈ ∂f ∗ (x∗ ). It is also worth pointing out that ∂f (x) 6= ∅ whenever the convex function f is finite at x and continuous at x (see, e.g., [6]). Given two convex functions f, g : Rn → R ∪ {+∞}, their infimal convolution f g given by  (f g)(x) := infn f (y) + g(x − y) for all x ∈ Rn , y∈R is a convex function, and clearly dom f g = dom f + dom g. When in addition one function is continuous at a point in dom f ∩ dom g, it is known (see, e.g., [6]) that f ∗ g ∗ is a proper lsc convex function and (f + g)∗ = f ∗ g ∗ and ∂(f + g)(x) = ∂f (x) + ∂g(x), ∀x ∈ Rn . (2) 2 Given a subset S ⊂ Rn , its indicator function δS : Rn → R∪{+∞} is defined by δS (x) = 0 if x ∈ S and δS (x) = +∞ if x ∈ Rn \ S. Then, the conjugate δS∗ coincides with the support function of S since δS∗ (x∗ ) = suphx∗ , xi for all x∗ ∈ Rn , x∈S so δS∗ (·) ≤ δS∗ ′ (·) if and only if cl co S ⊂ cl co S ′ . It is also worth noticing that, whenever S 6= ∅, the equality δS∗ (0) = 0 ensures that δS∗ is a proper lsc convex function. Further, dom δS∗ is clearly a convex cone containing zero. If S is a nonempty closed convex set, its recession cone 0+ S is defined by 0+ S = {y ∈ Rn : S + λ y ⊂ S, ∀λ ≥ 0}. The (negative) polar cone P o of a nonempty cone P ⊂ Rn is defined by P o := {u ∈ Rn : hx, ui ≤ 0, ∀x ∈ P }. Under the assumption that S is nonempty, closed and convex, it is known (cf, e.g. [6, Corollary 14.2.1]) that 0+ S = (dom δS∗ )o , (3) which can be also directly seen from the equality S = {x ∈ Rn : hx, x∗ i ≤ δS∗ (x∗ ), ∀x∗ ∈ dom δS∗ }. We recall now the concepts of Motzkin decomposable sets and functions. Definition 1 A nonempty closed convex set S in Rn is said to be Motzkin decomposable ( M-decomposable, for short) provided that there are a compact convex set K in Rn and a closed convex cone P in Rn such that S = K + P (the sum being taken in Minkowski’s sense). Such an equality is called a Motzkin representation (or decomposition) of S with compact and conic components K and P , respectively. A proper lsc convex function f : Rn → R ∪ {+∞} is then called Motzkin decomposable ( M-decomposable, for short) when its epigraph epi f is an M- decomposable set in Rn+1 . The following theorem recalls in another equivalent form the result of Propo- sition 16 in [1]; it will be crucial in the next section. Theorem 2 A nonempty closed convex set S in Rn is M-decomposable if and only if there exists a compact convex set K ⊂ S such that for each x∗ ∈ dom δS∗ there exists some x ∈ K such that δS∗ (x∗ ) = hx, x∗ i (in other words, dom δS∗ = ∂δS (K)). 3 3 Subdifferential characterization of M-decomposable functions Theorem 3 A proper lsc convex function f : Rn → R∪{+∞} is M-decomposable if and only if there exists a compact convex set K ⊂ dom f such that cl dom f = K +0+ (cl dom f ) (that is, K is a compact component of cl dom f ), f is bounded on K, and ∂f (K) = dom f ∗ . Proof. Assume first that f is M-decomposable, i.e., that epi f is an M-decomposable set. Then, by Theorem 2 there exists a compact set C ⊂ epi f such that for every (x∗ , α∗ ) ∈ dom δepi ∗ f there exists (x, α) ∈ C satisfying ∗ ∗ ∗ ∗ ∗ δepi f (x , α ) = hx, x i + αα . (4) b be the projection of C onto Rn , that is, Let C b := {x ∈ Rn : (x, α) ∈ C for some α ∈ R} . C b is a compact convex subset of dom f. We will now prove that Clearly, K := co C cl dom f = K + 0+ (cl dom f ) . Let x∗ ∈ dom δcl ∗ dom f . Since ∗ ∗ ∗ ∗ ∗ ∗ δepi f (x , 0) = δdom f (x ) = δcl dom f (x ) , we have (x∗ , 0) ∈ dom δepi ∗ f ; hence there exists (x, α) ∈ C satisfying hx, x∗ i = δepi ∗ ∗ ∗ ∗ f (x , 0) = δcl dom f (x ) . Since x ∈ C b ⊂ K, we deduce that δ ∗ ∗ ∗ ∗ cl dom f (x ) ≤ δK (x ). This being true ∗ ∗ ∗ for any x ∈ dom δcl dom f , it ensues by lower semicontinuity of δcl dom f and by ∗ n ∗ ∗ ∗ ∗ ∗ ∗ continuity of δK on R that δcl dom f (x ) ≤ δK (x ) for all x ∈ cl(dom δcl dom f ), ∗ ∗ ∗ which entails that cl(dom δcl dom f ) ⊂ dom δcl dom f . Then, dom δcl dom f is closed and ∗ ∗ ∗ ∗ ∗ ∗ δcl dom f (x ) ≤ δK (x ) ≤ δK+0+ (cl dom f ) (x ). o On the other hand, if x∗ 6∈ dom δcl∗ ∗ + dom f , we have x 6∈ (0 (cl dom f )) by (3), ∗ ∗ and hence δK+0+ (cl dom f ) (x ) = +∞ (as easily seen), so the latter inequality ∗ ∗ ∗ ∗ δcl dom f (x ) ≤ δK+0+ (cl dom f ) (x ) still holds true. We derive that cl dom f ⊂ K + 0+ (cl dom f ), and the inclusion is in fact an equality because K + 0+ (cl dom f ) ⊂ cl dom f + 0+ (cl dom f ) = cl dom f. This justifies the desired M-decomposability of cl dom f with K as compact component. We will next prove that f is bounded on K. Since f is lsc and K is compact, we only have to prove boundedness from above. Clearly, f is bounded above on 4 Cb by max {α ∈ R : (x, α) ∈ C for some x ∈ Rn } , and by the convexity of f the same upper bound applies on K. Let us now prove that ∂f (K) = dom f ∗ . Since the inclusion ⊂ obviously holds, we only need to prove the opposite inclusion. Let x∗ ∈ dom f ∗ . We have δepi ∗ ∗ ∗ ∗ ∗ ∗ f (x , −1) = f (x ) ; hence (x , −1) ∈ dom δepi f . Therefore, by (4) there exists (x, α) ∈ C satisfying hx, x∗ i − α = δ ∗epi f (x∗ , −1) = f ∗ (x∗ ) . Using the fact that α ≥ f (x) it readily follows that x∗ ∈ ∂f (x) ⊂ ∂f (K) , which proves the required inclusion, and hence the equality. The implication ⇒ is then justified. Conversely, let us now assume the existence of K as in the statement of the theorem, and define a := min {f (x) : x ∈ K} , b := sup {f (x) : x ∈ K} and C := K × [a, b] . The set C is obviously convex and compact. Let (x∗ , α∗ ) ∈ dom δepi ∗ f . Then ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ α ≤ 0. If α = 0 then, as δcl dom f (x ) = δepi f (x , 0) , we have x ∈ dom δcl dom f ⊂ ∗ dom δK ; hence there exists x ∈ K such that hx, x∗ i = δK ∗ (x∗ ) = δcl ∗ ∗ ∗ ∗ dom f (x ) = δepi f (x , 0) , where the second equality is due to the assumption cl dom f = K +0+ (cl dom f ) and to (3). We thus have ∗ ∗ ∗ δepi f (x , 0) = hx, x i + a0, and further (x, a) ∈ C by definition of C. Suppose now that α∗ < 0. Then, since  ∗   ∗  ∗ x ∗ x 1 ∗ f = δ epi f , −1 = δ (x∗ , α∗ ) , −α∗ −α∗ −α epi f x∗ we deduce that −α∗ ∈ dom f ∗ = ∂f (K), where the equality follows by assump-  ∗  D E x x∗ tion. Consequently, there is some x ∈ K such that f ∗ −α ∗ = x, −α∗ −f (x), and equivalently  ∗  ∗ ∗ ∗ ∗ ∗ x δepi f (x , α ) = −α f = hx, x∗ i + α∗ f (x) ; −α∗ moreover, (x, f (x)) ∈ C according to the definition of C. We can then apply Theorem 2 to conclude that epi f is M-decomposable, which means that f is an M-decomposable function. Remark 4 The ”if” statement does not require convexity of K; in fact, one can easily check that if a compact set K satisfies the stated conditions so does the compact convex set co K. A simple example where all the conditions in Theorem 3 are fulfilled with a nonconvex compact set K is furnished, as it is not difficult 5 to verify, by the compact set K := {−1, 1} in R and the function f : R → R defined by f (x) = max{|x|, 1} for all x ∈ R. Before stating a first corollary, let us establish the following lemma which has its own interest even in the setting of general normed spaces. Lemma 5 Let g : X → R ∪ {+∞} be a proper convex function on a normed space X and C be a subset of X for which there exists a bounded set Q in the topological dual space X ∗ such that ∂g(x) ∩ Q 6= ∅ for all x ∈ C. Then, the restriction of g to C is Lipschitz. Proof. By assumption there exist a real γ ≥ 0 with Q ⊂ γBX ∗ along with a mapping ζ : C → X ∗ such that ζ(x) ∈ ∂g(x) ∩ Q for all x ∈ C. Then, for any x, y ∈ C we have by definition of subdifferential of a convex function g(y) ≥ g(x) + hy − x, ζ(x)i ≥ g(x) − γky − xk, or equivalently g(x) − g(y) ≤ γkx − yk, which confirms the Lipschitz property of g on C. Corollary 6 If a proper lsc convex function f : Rn → R ∪ {+∞} is M- decomposable, then f ∗ is Lipschitz on its domain. Proof. Let K be the compact set in Rn given by Theorem 3. By this theorem and by (1) it easily follows that ∂f ∗ (x∗ ) ∩ K 6= ∅ for every x∗ ∈ dom f ∗ . Since K is bounded, this condition implies by the above lemma that f ∗ is Lipschitz on dom f ∗ . Remark 7 Consider the function f := k·k+δ{1}×R on R2 . For any (u, v) ∈ R2 we have by calculus rule for the Legendre-Fenchel conjugate of the sum of two convex functions one of which is continuous (see (2)),  f ∗ (u, v) = inf n δ{1}×R (x, y) + k(u, v) − (x, y)k (x,y)∈R = inf k(u, v) − (x, y)k = |u − 1|. (x,y)∈{1}×R (1,y) On the other hand, ∂f (1, y) = k(1,y)k + R × {0} for any y ∈ R (see the second equality in (2)), so there is no compact set K such that ∂f (K) = R2 . Since dom f ∗ = R2 , Theorem 3 ensures that f is not M-decomposable, while f ∗ is Lipschitz on its domain. This tells us that the converse of Corollary 6 fails. Despite the above counterexample, the situation is different when f is finite- valued on Rn . Corollary 8 Let f : Rn → R be a finite-valued convex function. Then the following statements are equivalent: 6 (a) f is M-decomposable. (b) There exists a compact convex set K ⊂ Rn such that ∂f (K) = dom f ∗ . (c) There exists a bounded set Q ⊂ Rn such that ∂f (Q) = dom f ∗ . (d) f ∗ is Lipschitz on its domain. Proof. The equivalence between (a) and (b) is an immediate consequence of Theorem 3. Implication (b) =⇒ (c) is trivial, and (c) implies (b) by taking K := cl co Q since by (1) and (c) we have dom f ∗ = ∂f (Q) ⊂ ∂f (K) ⊂ ∂f (Rn ) ⊂ dom f ∗ . Implication (a) =⇒ (d) follows from Corollary 6. It remains to prove that (d) implies (b). Assume that f ∗ is Lipschitz on its domain with some constant γ ≥ 0. Put K := γ B and note that ∂f (K) ⊂ dom f ∗ , since ∂f (Rn ) ⊂ dom f ∗ by (1). On the other hand, it is known (and not difficult to see) that the convex function ϕ := f ∗ (γ k · k) is finite-valued and γ-Lipschitz on Rn and it coincides with f ∗ on dom f ∗ . Fix any x∗ ∈ dom f ∗ . By continuity of ϕ we may choose some x ∈ ∂ϕ(x∗ ), and by the γ-Lipschitz property of ϕ we have kxk ≤ γ (as known and easily verified). Since ϕ ≤ f ∗ with ϕ(x∗ ) = f ∗ (x∗ ), the definition of subdifferential implies ∂ϕ(x∗ ) ⊂ ∂f ∗ (x∗ ), so x ∈ ∂f ∗ (x∗ ), or equivalently x∗ ∈ ∂f (x) according to (1). This ensures that dom f ∗ ⊂ ∂f (K), hence the equality ∂f (K) = dom f ∗ is shown, which finishes the proof of the corollary. 4 Sum and composition Consider on R2 the functions g := k · k and h := δ{1}×R . Corollary 8(d) tells us that g is M-decomposable since g ∗ = δB is Lipschitz on its domain, and the function h is also M-decomposable because its epigraph epi h = {1} × R × [0, +∞[= {(1, 0, 0)} + {0} × R × [0, +∞[ is an M-decomposable set. Nevertheless, the sum g + h is not M-decomposable according to Remark 7. The next proposition provides a case when the sum is M-decomposable. It has been previously established in [3, Theorem 25]. Our proof below is very simple. Proposition 9 Let f : Rn → R ∪ {+∞} be a proper lsc convex function, α ∈ R and ℓ : Rn → R be a linear function. If f is M-decomposable, then f + ℓ + α is also M-decomposable. Proof. The convex function ϕ := f +ℓ+α is clearly lsc and proper with dom ϕ = dom f . By Theorem 3 there exists a compact convex set K ⊂ Rn over which f is bounded and such that cl dom f = K + 0+ (cl dom f ) and ∂f (K) = dom f ∗ . 7 Obviously ϕ is also bounded on K and cl dom ϕ = K + 0+ (cl dom ϕ). Further, as easily seen ϕ∗ (x∗ ) = f ∗ (x∗ − y ∗ ) − α, with y ∗ ∈ Rn such that l = h·, y ∗ i , so dom ϕ∗ = y ∗ + dom f ∗ . From this and the above equality ∂f (K) = dom f ∗ (using the second equality in (2)) we see that ∂ϕ(K) = dom ϕ∗ . Altogether we obtain by Theorem 3 again that ϕ is M-decomposable. The case of the sum of two functions with separate variables is also of interest. Proposition 10 Let f : Rn → R ∪ {+∞} and g : Rm → R ∪ {+∞} be two proper lsc convex functions which are M-decomposable. Then, the function ϕ : Rn × Rm → R ∪ {+∞} defined by ϕ(x, y) := f (x) + g(y) for all (x, y) ∈ Rn × Rm , is M-decomposable. Proof. Suppose first that f and g are M-decomposable. By Theorem 3 there is a compact convex set Kf (resp. Kg ) in Rn (resp. Rm ) over which f (resp. g) is bounded and such that cl dom f = Kf + 0+ (cl dom f ) (resp. cl dom g = K + 0+ (cl dom g)) and ∂f (Kf ) = dom f ∗ (resp. ∂g(Kg ) = dom g ∗ ). The function ϕ is obviously proper, lcs and convex, and it is bounded on Kf × Kg . It is also easily checked from the equality dom ϕ = dom f × dom g that cl dom ϕ = Kf × Kg + 0+ (cl dom ϕ). Note also that ϕ∗ (x∗ , y ∗ ) = f ∗ (x∗ ) + g ∗ (y ∗ ), so dom ϕ∗ = (dom f ∗ ) × (dom g ∗ ). Then, since ∂ϕ(x, y) = ∂f (x) × ∂g(y) (as readily verified), we have ∂ϕ(Kf × Kg ) = ∂f (Kf ) × ∂g(Kg ) = (dom f ∗ ) × (dom g ∗ ) = dom ϕ∗ . Applying Theorem 3 again, we obtain that the function ϕ is M-decomposable. Concerning Cartesian products of sets we have the following result. Proposition 11 Let S and S ′ be two nonempty closed convex sets in Rn and Rm respectively. Then S × S ′ is M-decomposable in Rn × Rm if and only if S and S ′ are M-decomposable in Rn and Rm respectively. Proof. The implication ⇐ is obvious. Suppose that S × S ′ is M-decomposable in Rn × Rm and choose by definition a compact convex set K in Rn × Rm such that S × S ′ = K + 0+ (S × S ′ ). Denote by C (resp. C ′ ) the projection of K on Rn (resp. Rm ). The set C ×C ′ is convex and compact in Rn × Rm , and clearly C × C ′ ⊂ S × S ′ since K ⊂ S × S ′ . This and the obvious inclusion K ⊂ C × C ′ entail that S × S ′ = K + 0+ (S × S ′ ) ⊂ C × C ′ + 0+ (S × S ′ ) ⊂ S × S ′ + 0+ (S × S ′ ) = S × S ′ , 8 which combined with the (readily seen) equality 0+ (S × S ′ ) = (0+ S) × (0+ S ′ ) yields S × S ′ = C × C ′ + (0+ S) × (0+ S ′ ) = (C + 0+ S) × (C ′ + 0+ S ′ ). This gives S = C + 0+ S and S ′ = C ′ + 0+ S ′ , so S and S ′ are M-decomposable in Rn and Rm respectively as desired. Let us now turn to composition with linear mappings. Proposition 12 Let A : Rn → Rm be a linear mapping, S ⊂ A (Rn ) be a nonempty closed convex set and f : Rm → R ∪ {+∞} be a proper lsc convex function such that dom f ⊂ A (Rn ) . The following hold: (a) If the set S is M-decomposable, then A−1 (S) is M-decomposable. (b) If the function f is M-decomposable, then f ◦ A is also M-decomposable. Proof. (a) Let K be a compact convex set in Rm and P be a closed convex cone in Rm such that S = K + P . We claim that A−1 (S) = A−1 (K) + A−1 (P ). First, we note that, for any u ∈ A−1 (K) and v ∈ A−1 (P ), we have A(u + v) = Au + Av ∈ K + P = S, that is, u + v ∈ A−1 (S); this means that A−1 (K) + A−1 (P ) ⊂ A−1 (S). For the converse inclusion, fix any x ∈ A−1 (S). Then, Ax = y+z with y ∈ K and z ∈ P . By the inclusion S ⊂ A (Rn ) we have K ⊂ A (Rn ) and P ⊂ S − K ⊂ A (Rn ) , so there are u, v ∈ Rn such that y = Au and z = Av, hence Ax = Au + Av = A(u + v) with u ∈ A−1 (K) and v ∈ A−1 (P ). We thus have A (x − u) = Av ∈ P, which shows that x − u ∈ A−1 (P ), so A−1 (S) ⊂ A−1 (K) + A−1 (P ). The claim is then justified.  Let us now prove that A−1 (K) = A−1 (K) ∩ (Ker A)⊥ + Ker A. For any u ∈ A−1 (K) ∩ (Ker A)⊥ and v ∈ Ker A, we have A(u + v) = Au ∈ K, thus u + v ∈ A−1 (K). Conversely, let any x ∈ A−1 (K). Writing x = xA + x⊥ ⊥ with xA ∈ Ker A and x⊥ ∈ (Ker A) , we obtain K ∋ Ax = A(x⊥ ), hence x⊥ ∈ A−1 (K) ∩ (Ker A)⊥ . This gives x ∈ A−1 (K) ∩ (Ker A)⊥ + Ker A, so the equality  A−1 (K) = A−1 (K) ∩ (Ker A)⊥ + Ker A (5) + −1  −1  holds true. Further, since 0 A (K) ⊂ Ker A, the convex set A (K) ∩ (Ker A)⊥ is compact. To conclude that A−1 (S) is M-decomposable it suffices to note that A−1 (P ) is a closed convex cone and to write, by (5) and by the equality established above for A−1 (S), that   A−1 (S) = A−1 (K) ∩ (Ker A)⊥ + Ker A + A−1 (P )  = A−1 (K) ∩ (Ker A)⊥ + A−1 (P ), since Ker A + A−1 (P ) = A−1 (P ) by the inclusion Ker A ⊂ A−1 (P ). (b) Considering the linear mapping T : Rn × R → Rm × R defined by T (x, r) := (Ax, r), we see that epi (f ◦ A) = T −1 (epi f ). Therefore, (a) says that epi (f ◦ A) is M-decomposable, or equivalently the function f ◦ A is M-decomposable. 9 Remark 13 Without the assumption S ⊂ A (Rn ) , the assertion (a) above may fail. Let f : R2 → R and A : R3 → R3 be defined for all (x, y, z) ∈ R3 by p f (x, y) := (x − 1)2 + y 2 = k(x, y) − (1, 0)k and A(x, y, z) := (0, y, z), and let S := epi f. pConsidering the finite-valued convex function h : R → R defined by h(y) := 1 + yp 2 for all y ∈ R, by easy computation (or see [6, p. ∗ ∗ 106]) one has h (y ) = − 1 − (y ∗ )2 if |y ∗ | ≤ 1 and h∗ (y ∗ ) = +∞ otherwise, so h∗ is not Lipschitz on its domain. Corollary 8(d) tells us that h is not M- decomposable, i.e., the set epi h is not M-decomposable in R2 . Since A−1 (S) = R × epi h, invoking Proposition 11 it follows that the set A−1 (S) is not M- decomposable. Similarly, without the assumption dom f ⊂ A (Rn ), the assertion (b) above may fail. Let f be the function we have just considered and T : R2 → R2 be the linear mapping defined for all (x, y) ∈ R2 by T (x, y) := (0, y). Noting that f ◦ T coincides with the function h considered above, we see that f ◦ T is not M-decomposable while f is. Acknowledgments. J. E. Martı́nez-Legaz was partially supported by the MINECO of Spain, Grant MTM2014-59179- C2-2-P and the Severo Ochoa Pro- gramme for Centres of Excellence in R&D [SEV-2015-0563]. He is affiliated with MOVE (Markets, Organizations and Votes in Economics). References [1] M.A. Goberna, E. González, J.E. Martı́nez-Legaz, M.I. Todorov, Motzkin decomposition of closed convex sets, J. Math. Anal. Appl. 364 (2010), 209- 221. [2] M.A. Goberna, A.N. Iusem, J.E. Martı́nez-Legaz, M.I. Todorov, Motzkin decomposition of closed convex sets via truncation. J. Math. Anal. Appl. 400 (2013), 35–47. [3] M.A. Goberna, J.E. Martı́nez-Legaz, M.I. Todorov, On Motzkin decompos- able sets and functions, J. Math. Anal. Appl. 364 (2010), 525-537. [4] A.N. Iusem, J.E. Martı́nez-Legaz, M.I. Todorov, Motzkin predecomposable sets, J. Glob. Optim. 60 (2014), 635-647. [5] Th. Motzkin, Beiträge zur Theorie der Linearen Ungleichungen, Inaugural Dissertation 73 S., Basel, 1936. [6] R.T. Rockafellar, Convex Analysis, Princeton University Press, Princeton, 1970. 10